In this study we found that in absolute terms, the accessibility by car of recreational PA facilities in Scotland greatly exceeded that by bus, and that this difference was more pronounced for lower travel time thresholds where access and egress times constituted a greater proportion of total bus travel time. We also found that remote areas and, to a lesser extent, accessible rural areas had fewer opportunities to access PA facilities than urban areas and accessible small towns.
Our regression models showed that accessibility by car was significantly (p<0.01) higher for the most affluent quintile of area-based income deprivation than for most other quintiles in small towns and all other quintiles in rural areas, but this was not the case in urban areas. In contrast, access to PA facilities by bus for the most affluent quintile was significantly lower than that for other quintiles in urban areas and small towns, but not in rural areas. With the exception of access by car in urban areas and bus in rural areas, these results are consistent with the twin hypotheses that the most affluent areas enjoy the best access to PA facilities by car but the poorest access to these facilities by bus compared to other deprivation quintiles within each particular urban/rural class modelled in this paper.
When we sub-categorised PA facilities by ownership, we did not find any significant difference between private and public PA facility accessibility except for the lowest travel time threshold for bus in urban areas. It might be hypothesised that as a consequence of market forces private facilities are more likely to be located close to more affluent areas and that public policy would seek to redress this imbalance by locating public facilities in more deprived areas. Proximity by distance is closely related to access by bus within 10 minutes since our assessment of total journey time took into account access, egress and bus stop waiting times as well as in-vehicle time. Our results, which take into account the effect of population density, are consistent with this hypothesis to the extent that access to public facilities increased with increasing deprivation, whereas access to private facilities was significantly lower in the most deprived areas than in the more affluent quintiles.
In previous analysis we found that those living in ‘middling’ income areas enjoyed better access to PA facilities within the DZ of residence than those living in either low or high-income areas, after adjusting for population density, urbanicity and local authority 
. When we then examined the accessibility of facilities within walking and cycling distance we found, in general, that low and middling income areas experienced similar levels of accessibility and that the most affluent areas had poorer access than other areas 
. We also found that the accessibility gap between the most affluent and other areas became less pronounced with higher travel time thresholds and for cycling in comparison with walking. This shows that any apparent disadvantage experienced by the most deprived areas in comparison with middling income areas can be readily overcome within a relatively short travel time or distance. Furthermore, in this paper our results for car accessibility reveal that a journey of 10 minutes is sufficient to remove any differences in accessibility by area-based income deprivation in urban areas and to provide higher levels of accessibility in small towns and rural areas. This shows the degree to which the relative accessibility of PA facilities is sensitive to small changes in travel time threshold by walking and cycling or to choosing to travel by car for even short journeys.
With reference to the wider literature, to the best of our knowledge the only comparable research was that undertaken in metropolitan Perth, Western Australia which found that low socio-economic status (SES) areas enjoyed better road network access to PA facilities than high SES areas 
. Clearly, the apparent disparity between that work and our results on car accessibility may be attributable to the different geographical and social contexts of the two study areas. From a methodological perspective, a strength of our study was that it took into account the effect of road type and geometry-related junction delay on travel times, whereas the Australian study was based solely on road network distance. In addition, different approaches were employed in the determination of accessibility. Whereas we used the cumulative opportunities method, a gravity-based approach was used in the Australian study in which the closer a facility to an origin, the greater its contribution to the accessibility index. We found that differences in accessibility by area-based income deprivation increased when we reduced the travel time threshold and, at the lowest threshold examined (10 minutes), the most affluent areas enjoyed as good if not better access by car, a finding which is apparently at odds with the results of the Australian study. However, as discussed in the previous paragraph, if we were to reduce the car travel time threshold to below 10 minutes (or, alternatively, by considering walk/cycle accessibility), we would anticipate a steeper decline in accessibility in the most affluent areas than elsewhere. This suggests that at least some of the difference between the results of the two studies may be attributable to the method used to calculate accessibility, but in the absence of behavioural data to estimate the deterrent effect of travel time (for different modes) in our study area, it is not possible to pursue this question further using currently available data.
It is clear from our analysis that car ownership is an important factor in determining the level of accessibility to PA facilities enjoyed by an individual. We have seen in that there is a considerable variation in the proportion of households without access to a car in Scotland and that a large proportion of Scottish households must rely on non-car modes of transport to use PA facilities. Our results for bus accessibility in urban areas and in small towns showed that the most affluent quintile had a lower level of accessibility than other quintiles. This suggests that bus services are weighted in favour of less well-off areas and that this helps to connect the population of these areas with PA facilities. At the same time, the groups which are most disadvantaged according to our results are those without access to a car and living in the most affluent areas or in rural areas.
An obvious extension to our analysis would be to explore the intersection of car ownership with accessibility by income deprivation and URC. Car ownership data is available for households at DZ level in Scotland but the amount of information on the size and structure of households presented with this data is limited. This means that e.g. the number of cars per adult household member is not known. There is also the added difficulty of interpreting the results since individual household members are likely to have unequal access to any cars owned by their household as a result of driving licences held, work commuting and other such constraints. Nonetheless, we performed a tentative analysis in which we calculated a compound accessibility score for each DZ by weighting car and bus population adjusted accessibilities by the proportion of households with and without a car respectively. We then calculated the median compound accessibility for each travel time threshold and income deprivation quintile nested within urban, small town and rural categories. The full results are shown in Table S3
. Despite higher car ownership in rural areas, there remained a clear advantage in compound accessibility for urban areas over rural areas. As expected given the positive association between income and car ownership and the gap between car and bus accessibilities, the most affluent quintile enjoyed the highest level of compound accessibility. In urban areas, compound accessibility declined with increasing deprivation for all travel time thresholds. In small towns and rural areas this was also the case for the 10 minute travel time threshold, although no trend was apparent for the 20 and 30 minute thresholds.
This paper is one of the few to have examined the accessibility of PA facilities beyond the level of the neighbourhood and is the first to develop a GIS model of car and bus networks for this purpose. One limitation of the model was that it did not take into account the effect of congestion-related delays on car travel times. Another limitation was that we assumed a constant speed for access to and egress from bus services which did not take into account the effect of the roadway environment on such speeds. Likewise, the effect of bus service frequency on accessibility was not fully accounted for. As described in the methods section, bus stop waiting times were estimated to be a function of the service headway up to a maximum of five minutes, based on the assumption that passengers would have prior knowledge of the timetable and so would schedule their journey accordingly. However, this does not take into account any underutilised time at home or at the PA facility, which would tend to be greater for routes with less frequent services, nor does it capture the effect of constraints placed by (or upon) other activities on journey planning.
With the exception of disaggregating the PA facility data set along public/private ownership lines, we took no account of potential preferences for particular types of facility in our analysis. We know from population health surveys in Great Britain that preferences for different forms of PA vary by age and gender 
. It is also reasonable to infer that the attractiveness of different forms of sport/exercise is socially and culturally dependent. We have partly addressed the issue of varying potential preferences for different types of PA facility in a companion paper in which we classified facilities by the intensity of PA offered 
. With a suitable data set it would also be possible to examine differential access to facilities with specific attributes such as the availability of specialist coaching and reserved sessions for particular demographic groups. It would also be appropriate to widen the scope of the research to include opportunities for PA in informal settings.
The cumulative opportunities accessibility measure used in this research reveals the total number of PA facilities available to an individual within a specified travel time. We examined the accessibility gradient across three travel time thresholds. Our approach did not take into account the distance or time individuals were prepared to travel to reach PA facilities. Recent evidence from Australia revealed that the average network distance travelled to formal recreational facilities was around 5.5km 
. In future research, therefore, it would be useful to explore how far individuals are prepared to travel to undertake PA in a Scottish context, and how this varies by mode and by sex, gender and other socio-economic characteristics. Furthermore, it is recognised that the value of accessibility above a certain satisfactory and sufficient level depends on the degree, or the diversity, of choice available 
. This means that, for example, if a community had two accessible swimming pools but no facilities for tennis, the provision of a tennis facility would tend to add more value to the community than the provision of an additional swimming pool. This “value” may arise in satisfying a taste for variety in PA (e.g. using facilities with different attributes, taking part in different forms of PA) or in providing opportunities for novices to try out different forms of sport/exercise. The provision of an appropriate mix of facilities is also important in satisfying different preferences at household or community level as discussed in the preceding paragraph. Alternative formulations of accessibility have been proposed in the literature to capture diversity (see, for example, 
) and there is clearly scope to extend our research in this direction.
The Scottish Government recognises the link between poor health and deprivation 
and it views increasing physical activity as an important component of its policy to tackle ill-health 
. Furthermore, increasing participation in sport is regarded as an important legacy of the 2012 London Olympic Games 
and the XX Commonwealth Games in Scotland 
. The aim of this study was to examine how accessibility to PA facilities by car and bus varies by area deprivation and thereby identify which groups are least well-served by the provision of PA facilities. We have shown that, when considering urban areas and small towns, residents of the most affluent areas without access to a car have the least good access to PA facilities. ( shows that a non-trivial number of households in the most affluent areas of Scotland do not own a car). This highlights the need for planners to consider all those without access to a car, not simply those living in more deprived areas, when determining the number and location of public PA facilities and in securing commitments towards local PA facilities from developers of new housing schemes. It has particular implications for the construction of affordable housing within larger housing developments on the periphery of existing settlements where high quality public transport is less likely to be commercially viable. This means that households without access to a car would be better off (in terms of access to PA facilities) if they were to live in more deprived areas. Even for low-income households with access to a car and living in the most affluent areas, the proportion of income spent on transportation tends to be particularly high 
. Part of the solution, therefore, lies in ensuring that public policy encourages investment in local PA facilities where possible. This would also have the effect of integrating and aligning policies on sports participation and reduction in carbon emissions from motorised transport. Where this is not possible, for reasons of finance, low rates of demand, lack of available land and suchlike, consideration should be given to clustering PA facilities in areas of high public transport accessibility. Measures to support cycling which provides relatively inexpensive access to mid-range destinations within a reasonable travel time would also be valuable. The picture in remote and rural areas is different in that there is no clear trend in accessibility of PA facilities with increasing deprivation. Although relevant, the policy measures discussed above are likely to play a less important role because either they are more difficult to implement or are less effective largely as a result of lower population densities. One of the key problems across parts of Scotland is the lack of access to a regular bus service. In these situations, provision of demand responsive transport offering greater flexibility and wider network coverage is an important part of the solution